Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3516⋊C3 |
| Fitting, Sylow | #3516 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3516 | C3 |
| hypercenter | #123 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #123 | |
267 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(10,14)(11,12,17)(15,22,23,21,16,24)(18,20,19);(2,5)(6,8)(9,17,13,19)(10,23,14,24)(11,20,18,12)(15,22,21,16);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)Polycyclic presentation:
2^3,3,2^4|a^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=g,g^d=fg,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3516⋊C3, #62788⋊C3, #62810⋊C3, … |
| maximal normal subgroup of | C2×#3516⋊C3, #62788⋊C3, #62810⋊C3, … |
| maximal quotient of | #34192⋊C3, #34262⋊C3, #34263⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18192);